Subnormal-to-normal properties

General remarks

Most of the properties we discuss here satisfy the intermediate subgroup condition. A subgroup property satisfies the intermediate subgroup condition if whenever are such that satisfies in , also satisfies in .

Most of the properties here are not closed under intersections. Some are closed under joins, and many are closed under normalizing joins. A subgroup property is normalizing join-closed if whenever are such that both satisfy and , the join also satisfies .

We will follow a right-action convention to denote conjugation. Thus, , will be termed the conjugate of by . In the left-action convention, is the conjugate of by . To use the left-action convention, replace by and reverse the order of terms in products.

Pronormal subgroup

A subgroup of a group is termed pronormal in if, for any , there exists such that .

Any normal subgroup is pronormal, and any maximal subgroup is pronormal. More generally, there are several properties between the property of being normal or maximal and the property of being pronormal. Here are some of them:

Pronormal implies WNSCDIN. Pronormal implies MWNSCDIN: Pronormal subgroups satisfy a strong condition on conjugacy of normal subsets. Any two normal subsets of a pronormal subgroup that are conjugate in the whole group are conjugate in the normalizer of the pronormal subgroup.

Weakly pronormal subgroup

The condition of being a weakly pronormal subgroup is a slight weakening of the condition of being a pronormal subgroup. Here, instead of looking at the subgroup generated by and one conjugate , we look at the subgroup generated by and the conjugates of by all powers of . This subgroup is denoted . We require that there exist such that .

Paranormal subgroup

Paranormality is a weakening of pronormality, but in a somewhat different direction. For pronormality, we insist that there exist such that . For paranormality, we weaken this from requiring the existence of a single element to simply requiring that be a contranormal subgroup inside : in other words, we require that the normal closure of in be .

NE-subgroup

The condition of being a NE-subgroup is stronger than weak normality, but is incomparable with paranormality, pronormality, or weak pronormality. Every normal subgroup as well as every self-normalizing subgroup (we'll see more on those later) is a NE-subgroup.

Intermediately subnormal-to-normal subgroup

A subgroup of a group is termed intermediately subnormal-to-normal in if whenever is a subgroup of containing , and is a subnormal subgroup of , is also a normal subgroup of . This is equivalent to demanding that whenever is a 2-subnormal subgroup of , is normal in .

All the properties mentioned above are stronger than the property of being intermediately subnormal-to-normal.

Note: The concept of intermediately subnormal-to-normal was referred to in one paper as transitively normal while the concept of image-closed intermediately subnormal-to-normal was referred to in the same paper as strong transitively normal.

Normal-to-characteristic and normal-to-isomorph-free properties

General remarks

For this part, we shall follow the convention of automorphisms acting on the right, by exponentiation. This is to keep consistent with the notation of conjugation on the right. To convert these to analogous statements with automorphisms acting on the left, simply interchange the order of terms in all products.

Some of these properties have this behavior in intermediate subgroups as well: whenever a subgroup with the property is normal in an intermediate subgroup, it is characteristic (or isomorph-free) in that intermediate subgroup.

Isomorph-conjugate and automorph-conjugate subgroups

We say that is isomorph-conjugate in if any subgroup of isomorphic to is conjugate to . Similarly, we say that is automorph-conjugate in if any subgroup of that is automorphic to (i.e., is the image of under an automorphism of ) is conjugate to .

We have the following:

Neither the property of being isomorph-conjugate nor the property of being automorph-conjugate satisfy the intermediate subgroup condition. In other words, we can have such that is isomorph-conjugate in but not in .

Notice that, since any normal subgroup that is automorph-conjugate (resp., isomorph-conjugate) is in fact characteristic (resp., isomorph-free), each of these properties is stronger than the property of being an intermediately normal-to-characteristic subgroup.

Procharacteristic and weakly procharacteristic

Procharacteristicity is something like being automorph-conjugate in every intermediate subgroup, except that it is somewhat different. For the definition of procharacteristicity, we require that the automorphism be only in the ambient group, while conjugacy be checked in intermediate subgroups. Specifically:

A subgroup of a group is termed procharacteristic if, for any automorphism of , there exists such that .

One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.

Weakly characteristic and intermediately weakly characteristic

One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.

Other properties

There are many other normal-to-characteristic properties of interest. Some of these are discussed below:

Relating subnormal-to-normal and normal-to-characteristic

The general form of the relation

The properties we discussed in the last two sections: the sort that help us go from subnormal to normal, and the sort that help us go from normal to characteristic, are closely related. They are related by means of the left residual, something we shall try to describe here.

In our case, we observe that the left residual of a subnormal-to-normal property by the property of being a normal subgroup is a normal-to-characteristic property. If is the subnormal-to-normal property and is the normal-to-characteristic property, our general results will be of the form:

Any subgroup with property in a normal subgroup has property in the whole group.

If is a subgroup of such that whenever is normal in , is also normal in , has property in .

To understand why this is so, observe that the properties at the bottom here are all obtained as left residuals of certain properties by normality, each of which is stronger than the property of being an intermediately subnormal-to-normal subgroup. In particular, a subgroup of a normal subgroup with any of these properties inside the normal subgroup, is intermediately subnormal-to-normal in the whole group.

Hall subgroups need not be isomorph-conjugate or automorph-conjugate. However, Hall implies join of Sylow subgroups, and thus, Hall subgroups are paracharacteristic subgroups, and in particular, any Hall subgroup of a normal subgroup is paranormal. This allows us to prove a number of things about Hall subgroups as well.

Contranormality is UL-join-closed: Suppose, for some indexing set , we have subgroups , with , such that each is contranormal in . Then, the join of the s is contranormal in the join of the s.

It is true that the only subgroup that is both normal and contranormal is the whole group. In fact, the only subgroup that is both subnormal and contranormal is the whole group. This is because given any proper subnormal subgroup , there is a subnormal series:

.

The right-most member of this series that is not equal to is a proper normal subgroup of containing . Hence, cannot be contranormal.

Abnormal subgroup

The condition of abnormality is somewhat stronger than the condition of weak abnormality. is abnormal in if, for every , we have . Abnormality corresponds to pronormality just as weak abnormality corresponds to weak pronormality. An abnormal subgroup is precisely the same as a self-normalizing pronormal subgroup. Also, normalizer of pronormal implies abnormal.